Spin–orbit coupling in quantum materials: emergent phenomena, their modelling and examples from two-dimensional magnets

Abstract Acknowledging spin–orbit coupling (SOC) as a pivotal microscopic ingredient in quantum materials, this paper provides a comprehensive overview of spin–orbit-driven phenomena and effects in magnetic, non-magnetic and multiferroic materials. As for magnets, the review discusses SOC-induced magnetic anisotropy, exotic spin–spin interactions, and their implications on the emergence of complex spin textures. It also explores band-splitting effects in non-magnetic solids, with a focus on Rashba effects and spin-valley coupling. Additionally, the emergence of ferroelectric polarization from non-collinear spin textures in multiferroics is examined. The paper analyzes methods for estimating SOC-related quantities from first principles within the density functional theory, particularly exchange coupling tensors and magnetoelectric coupling tensors. Finally, it presents case studies on two-dimensional magnetic materials, including the characterization of peculiar Kitaev-like exchange coupling and the investigation of multiferroicity in NiI $$_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mn>2</mml:mn> <mml:mrow/> </mml:mmultiscripts> </mml:math> monolayers. Overall, the paper delves into the microscopic mechanisms related to SOC, by offering insights into the diverse effects and manifestations of SOC in the quantum materials world.